Borders between the Maghreb region's two most populous nations have been closed since 1994, despite repeated calls by Morocco in recent years for their reopening.
Algeria unilaterally cut ties with Morocco in 2021 and halted the flow of a gas pipeline to Spain via Morocco. It later banned all Moroccan aircraft from its airspace.
The Western Sahara dispute has been at the heart of worsening ties between the two countries.
Morocco considers the territory its own, but the Algerian-backed Polisario Front wants to establish an independent state there.
“I pray to Almighty God for things to return to normality, and for the opening of borders between the two neighbouring, sister countries and peoples,” said Morocco's King in a speech.
He also welcomed Israeli recognition of Morocco's sovereignty over Western Sahara, a decision fiercely criticised by Algeria, which has previously expressed discontent at the growing security ties between Rabat and Tel Aviv.
“I should like to tell the leaders and people of our sister nation, Algeria, that no evil will ever be done to them, nor will any harm ever come to them from Morocco,” King Mohammed said.
Algeria's President Abdelmadjid Tebboune said earlier this year that relations with Morocco had reached “the point of no return”.